How can I animate a Mobius strip on local Z axis?
Have you tried adding a driver to your z rotation axis? I would start by adding a driver on the z- rotation of your mobius strip. Create an empty object, and animate it going up on the z-axis or something, and use that as your drivers input variable. This should work if you only need to rotate on the z-axis.

Here's a video for a quick look at drivers: 
enter link description here

Aaand here's the docs if you really want to dive in:
Blender Driver Docs

I took your mobius blend file and applied this method:



Hope this helps!